# Jordan triple system

A triple system closely related to Jordan algebras.

A triple system is a vector space over a field together with a -trilinear mapping , called a triple product and usually denoted by (sometimes dropping the commas).

It is said to be a Jordan triple system if

(a1) |

(a2) |

with .

From the algebraic viewpoint, a Jordan triple system is a Lie triple system with respect to the new triple product

This implies that all simple Lie algebras over an algebraically closed field of characteristic zero, except , and (cf. also Lie algebra), can be constructed using the standard embedding Lie algebra associated with a Lie triple system via a Lie triple system.

From the geometrical viewpoint there is, for example, a correspondence between symmetric -spaces and compact Jordan triple systems [a3] as well as a correspondence between bounded symmetric domains and Hermitian Jordan triple systems [a2].

For superversions of this triple system, see [a5].

## Examples.

Let be an associative algebra over (cf. also Associative rings and algebras) and set , the -matrices over . This vector space is a Jordan triple system with respect to the product

where denotes the transpose matrix of .

Let be a vector space over equipped with a symmetric bilinear form . Then is a Jordan triple system with respect to the product

Let be a commutative Jordan algebra. Then is a Jordan triple system with respect to the product

Note that a triple system in this sense is completely different from, e.g., the combinatorial notion of a Steiner triple system (cf. also Steiner system).

#### References

[a1] | N. Jacobson, "Lie and Jordan triple systems" Amer. J. Math. , 71 (1949) pp. 149–170 |

[a2] | W. Kaup, "Hermitian Jordan triple systems and the automorphisms of bounded symmetric domains" , Non Associative Algebra and Its Applications (Oviedo, 1993) , Kluwer Acad. Publ. (1994) pp. 204–214 |

[a3] | O. Loos, "Jordan triple systems, -symmetric spaces, and bounded symmetric domains" Bull. Amer. Math. Soc. , 77 (1971) pp. 558–561 |

[a4] | E. Nehr, "Jordan triple systems by the graid approach" , Lecture Notes in Mathematics , 1280 , Springer (1987) |

[a5] | S. Okubo, N. Kamiya, "Jordan–Lie super algebra and Jordan–Lie triple system" J. Algebra , 198 : 2 (1997) pp. 388–411 |

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Jordan triple system.

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